Nessyahu–Tadmor-type central finite volume methods without predictor for 3D Cartesian and unstructured tetrahedral grids

Arminjon, Paul; St-Cyr, Amik

doi:10.1016/s0168-9274(03)00025-4

Cited by 40 publications

(30 citation statements)

References 27 publications

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“…With badly shaped tetrahedra (slivers) which are nearly flat and exhibit large dihedral angles (Figure 15), these problematic cells create an additional difficulty for the capture of the shock and particularly of the contact discontinuity. In [4,6,8] we have shown that our scheme is second-order for good shaped tetrahedra (isotropic elements) and also for 2D [27]. A fully converged steady-state solution was achieved in 6,000 iterations.…”

Section: Shock Tube Problem (Global Refinement)mentioning

confidence: 82%

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Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

Madrane¹

2008

Hyperbolic Problems: Theory, Numerics, Applications

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Abstract.We present an explicit second-order finite volume generalization of the onedimensional (1D) Nessyahu-Tadmor schemes for hyperbolic equations on adaptive unstructured tetrahedral grids. The nonoscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is bypassed thanks to the use of the staggered Lax-Friedrichs scheme, is extended here to a two-steps scheme. In order to reduce artificial viscosity, we start with an adaptively refined primal grid in three dimensions (3D), where the theoretical a posteriori result of the first-order scheme is used to derive appropriate refinement indicators. We apply those methods to solve Euler's equations. Numerical experimental tests on classical problems are obtained by our method and by the computational fluid dynamics software Fluent. These tests include results for the 3D Euler system (shock tube problem) and flow around an NACA0012 airfoil. 1. Introduction. The history of schemes on staggered grids can at least be traced back to the famous paper of Courant, Friedrichs, and Lewy in 1928 [13] in which they discovered a scheme on staggered grids for the linear wave equation in one-dimensional (1D). For a special system arising in fluid dynamic problems von Neumann and Richtmyer used staggered grids as well [38]. Four years later, Lax introduced the well-known Lax-Friedrichs scheme and analyzed it [29]. In 1990 Tadmor and Nessyahu [41] picked up the idea to use staggered grids, showed the connection to Godunov's method, and proposed a second-order extension to 1D systems.The main advantage of these schemes is that no information about solutions to local Riemann problems is needed. Using staggered grids one can replace the upwind fluxes by central differences. The price one has to pay is the occurrence of excessive numerical viscosity since the resulting scheme can be interpreted as a Lax-Friedrichs scheme. Therefore, a higher order scheme of monotone upstream-centered schemes for conservation laws (MUSCL)-type in one spatial dimension was proposed in [41]. Later in [4,3,6,7,8,36] central schemes were generalized to multidimensional schemes on unstructured grids. For a Cartesian grid, we refer to [9,22,30,28,31,39,40,45,32] for related work.In the case of staggered unstructured multidimensional grids, there exist only a few convergence results. In [5] convergence of a second-order central scheme on twodimensional (2D) grids has been proven for a linear conservation law. Convergence of the first-order Lax-Friedrichs scheme on the same staggered grids for nonlinear scalar problems has been proven in [19].

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Section: Shock Tube Problem (Global Refinement)mentioning

confidence: 82%

“…Therefore, a higher order scheme of monotone upstream-centered schemes for conservation laws (MUSCL)-type in one spatial dimension was proposed in [41]. Later in [4,3,6,7,8,36] central schemes were generalized to multidimensional schemes on unstructured grids. For a Cartesian grid, we refer to [9,22,30,28,31,39,40,45,32] for related work.…”

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confidence: 99%

Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

Madrane¹

2008

Hyperbolic Problems: Theory, Numerics, Applications

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“…The scheme proposed in [30] is used by Bryson and Levy [31] to solve shallow-water equations with slope source terms. A modification of the NessyahuTadmor [14] scheme for hyperbolic conservation laws is presented by Arminjon and St.Cyr [32] and is adapted to unstructured triangular grids.…”

Section: Introductionmentioning

confidence: 99%

Central WENO scheme for the integral form of contravariant shallow‐water equations

Gallerano

Cannata

2010

Numerical Methods in Fluids

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A new Central Weighted Essentially Non-Oscillatory scheme for the solution of the shallow-water equations expressed in contravariant formulation is presented. One of the most efficient methodologies belonging to Central WENO family involves: reconstructions of cell-averaged values of flow variables, reconstruction of point-values of flow variables, advancing from time level t(n) to time level t(n+1) of the cell-averaged values. The extension of the above-mentioned methodology into the contravariant environment implies that the contravariant shallow-water equations must be expressed in an integral form. An element of novelty presented in this paper regards the definition of a formal integral expression of the shallow-water equations in a contravariant formulation, in which the Christoffel symbols are avoided. The WENO reconstructions are performed by a two-dimensional interpolating procedure taking into account the curved coordinate lines. The proposed scheme ensures the satisfaction of the exact C-property. Several two-dimensional test cases are used to verify the good resolution for smooth and discontinuous solutions. Copyright (C) 2010 John Wiley & Sons, Ltd

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“…High resolution generalizations of the NT scheme were developed since the 90s as the class of central schemes in e.g. [43,3,22,21,36,6,25,2,27,28,32], and here too, the list is far from being complete. The relaxation scheme of Jin and Xin [23] provides another approach which leads to a staggered central stencil for solving nonlinear conservation laws.…”

Section: Introductionmentioning

confidence: 99%

Central Discontinuous Galerkin Methods on Overlapping Cells with a Nonoscillatory Hierarchical Reconstruction

Liu¹,

Shu²,

Tadmor³

et al. 2007

SIAM J. Numer. Anal.

106

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Abstract. The central scheme of Nessyahu and Tadmor [J. Comput. Phys, 87 (1990)] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys, 160 (2000)] employs a variable control volume, which in turn yields a semi-discrete non-staggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu; J. Comput. Phys, 209 (2005)]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunov-type finite volume schemes. Overlapping cells lend themselves to the development of a central-type discontinuous Galerkin (DG) method, following the series of work by Cockburn and Shu [J. Comput. Phys. 141 (1998)] and the references therein.In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities, beyond those of Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which removes spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine and Flaherty [Appl. Numer. Math. 14 (1994)], but is otherwise different in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multi-layer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes and retains the overall pre-processed order of accuracy while effectively removes spurious oscillations around shocks.

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Nessyahu–Tadmor-type central finite volume methods without predictor for 3D Cartesian and unstructured tetrahedral grids

Cited by 40 publications

References 27 publications

Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

Three-Dimensional Adaptive Central Schemes on Unstructured Staggered Grids

Central WENO scheme for the integral form of contravariant shallow‐water equations

Central Discontinuous Galerkin Methods on Overlapping Cells with a Nonoscillatory Hierarchical Reconstruction

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